Geometry8.7 Relation Between the Sides of a Triangle and T-Ratios

The Law of sines
In any triangle, the sides are proportional to the sines of the opposite angles, i.e.,
a/sin A = b/sin B = c.sin CProof: Let ABC be any oblique triangle. Draw the altitude CD = h. Two cases are possible; all angles are acute or one angle obtuse.
In either case;
Sin B = h/a from the first figure
h = a sin B
From the second figure
sin A = h/b
h = b sin A
Equating the two values of h we get, asin B = bsin A => a/sin A = b/sin B
Similarly by drawing perpendicular from A to BC, we can prove that
c/sin C = b/sin B
Therefore in any triangle a/sin A = b/sin B = c/sin C